The recognition of specific signatures (i.e, patterns) in images and signals has long been of interest. Powerful techniques exist for their detection and classification, but these techniques are often defeated by changes or variations in the signature. These variations often include translation and scale changes. Methods exist for transforming the signal/image so that the result is invariant to these disturbances. Translation and scaling are well understood in a mathematical sense, so it is fairly straightforward to design methods which yield a transformed form of the data wherein these effects are removed. There are other variations which are not well described mathematically or are not mathematically tractable in terms of reasonable transformations.
Time-frequency (t-f) analysis is useful for signals which exhibit changes in frequency content with time. The well known spectrogram often presents serious difficulties when it is used to analyze rapidly varying signals, however. If the analysis window is made short enough capture rapid changes in the signal, it becomes impossible to resolve signal components which are close in frequency with the analysis window duration. If the window is made long to resolve the frequencies of sinusoids, the time when sinusoidal components act becomes difficult to determine.
Until recently, there was one alternative t-f analysis technique which was widely believed to avoid some of the problems of the spectrogram. The well known Wigner distribution (WD) avoids the problems of windowing and enjoys many useful properties, but often produces an unacceptable amount of interference or cross-term activity between signal components when the signal consists of many components. Despite its shortcomings, the Wigner distribution has been employed as an alternative to overcome the resolution shortcomings of the spectrogram. It also provides a high resolution representation in time and in frequency. The WD has many important and interesting properties.
Both the spectrogram and the WD are members of Cohen's Class of Distributions. Cohen has provided a consistent set of definitions for a desirable set of t-f distributions which have been of great value in this area of research. Different members of Cohen's class can be obtained by using different kernels. In this framework, the WD has a unity valued kernel. Choi and Williams introduced the Exponential Distribution (ED), with kernel EQU .PHI.ED(.theta.,.tau.)=e.sup.-.theta..sup..sup.2 .sup..tau..sup..sup.2 /.sigma.,
where .sigma. is a kernel parameter (.sigma.&gt;0). The ED overcomes several drawback of the spectrogram and WD providing high resolution with suppressed interferences.
The Reduced Interference Distribution (RID), which is a generalization of the ED, shares many of the desirable properties of the WD, but also has the important reduced interference property. RID is disclosed in U.S. Pat. No. 5,412,589, which also discloses a design procedure for Reduced Interference Distribution (RID) kernels. Generally, one may start with a primitive function, h(t), which has certain simple constraints, and evolve a full-fledged RID kernel from it. The RID kernel retains a unity value along the .theta. and .tau. axes in the ambiguity plane, generally providing good time-frequency resolution and auto-term preservation, but attenuates strongly elsewhere for good cross-term suppression.
Time-Frequency
The Wigner distribution has aroused much interest in the signal processing community. However, its use in image processing has been limited. Jacobson and Wechsler have pioneered in the use of such techniques in image processing. Cristobal et al. have investigated the use of Wigner distributions in image pattern recognition. Jacobson and Wechsler apparently have had a keen interest in human perception and the means by which images are perceived.
A more recent paper by Reed and Wechsler discusses the use of Wigner-based techniques to realize the Gestalt laws that resulted from perceptual grouping in the 1920s. It was suggested at this time that individual elements appear to group according to a set of principles including proximity, similarity, good continuation, symmetry and common fate. Reed and Wechsler go on to show that applying a relaxation procedure to the primary frequency plane of the 2D Wigner distribution is useful. Selection of the primary frequency plane reduces the representation from a N.times.N.times.N.times.N representation to a N.times.N frequency representation, of the same dimension as the original image. This is achieved by selecting pixels according to their maximum energies and retaining a number of top-ranked frequencies. Then, regions of homogeneity are grouped together. They also show that this process produces a similar end result for image textures that have the various Gestalt properties in common. This work is interesting and deserves further attention. One may conclude that the surface has been barely scratched in the application of space-spatial frequency techniques to images in general.